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Unformatted text preview: b u +ICJ/‘LJ Class Test 1 ECE502 — Analysis of Probabilistic Signals and Systems (Fall 2009)
Instructor: Professor A. M. Wygljnski October 5, 2009
06:00pm — 06:50pm Instruct ions 0 The duration of this class test is 50 minutes. 0 This class test consists of three (3) questions. each of which has a different point value.
a This class test is closed book. a One letter size sheet with handwritten notes on both sides is permitted. a A photocopy of the inner cover of the course textbook is permitted. 9 Calculators are permitted. 0 Please write neatlyr and legibly.  Each point in this class test corresponds to one minute of doing the test. Therefore, this
test is out. of 50 points. Budget. your time accordingly. : Attempt all questions: clearly show all your work and not just the ﬁnal answer.
 Collaborating with other students during the class test is strictly prohibited.
Hand in all material to the instructor at the end of 50 minutes. I (v'oori luck! m/so :~) WW Last Name First Name .Student ID Mailbox Question 1: Multiple Random Variables [25 Points] In H. Mass of ‘20 students, suppose that the number of coins in each student’s pocket can be deﬁned
as a uniform random variable between zero and twenty five. Asstune that the number of coins in the
packets of the different students is independent. (21) 4 points: Deﬁning the mnnber of coins centajned in the pocket of student k as the random
Variable Xk, express the probability P({X1 Z n} ﬂ'({X2 2 71} ﬂ    ﬂ{Xm Z n}) in terms of
P(,\'i.2n),k:1.21.”,20. (b) 4 points: Find the probability that no student has fewer than ﬁve coins in his/her pocket. HINT:
Use the result in part (a). (c) 5 points: What it the probability that at least one student has at least 19 coins in his / her pocket? ((1) 6 points: W hat is the probability that only student 1:: has catectly 19 coins in his/her pocket? (e) 6 points: What. is the probability that only one student has exactly 19 coins in his/her pocket?
HINT: Use the result from pent and generalize. Question 1 (Continued) Question 1 (Continued) Question 2: Conditional Probability [15 Points] The Acme Probability Products Company has been tasked to construct a random number generator
for a client. The client specifically requests a random number generator possessing an output statisti
cally deﬁned as a Poisson probability mass function (pnﬁ). Moreover, the client wants this generator
to be capable of using one of three possible A parameters: a, b, or (2. Consequently. the Acme Probability Products Company proposed the design shown in Figure l,
where Y is the sum of the random outputs of a bank of random number generators. and X is used
to select the desired random number generator. When a random number generator is not selected, it
only generates a steady stream of zeros. Thus, when X = 1 the output Y ~ Poisson (a), when X : 2
the outth i" ~ Poisson(b), and when X = 3 the output Y «1 Poisson(c). Note that X consists only
of the equally likely values 1. 2, and 3. < i ___.Flk
t.
v *o i
Q. E
a: I
3 i
E. a
e i _.—si: : 3 Poisson(c) __ ates Switch Bank of Random
Number Generators Figure 1: Schematic of a random number generator bank employing a selector switch. (a) 3 points: What are the mathematical expressions for the conditional probabilities P(Y = an =
1). my = nX = 2), and PO” = 10le = 3)? (b) 6 points: What is the probability that the output Y is a value less than or equal to 1, Le...
PO" 5 1)? (c) 6 points: What is the probability that the input to the selector switch is equal to 2 given that
th. ' is a value less than or equal to 1, 216., P[X = 2Y 5 1)? Question 2 (Continued) Question 3: Moment Generating Functions [10 Points] Suppose the moment. generating function of a, random variable X is deﬁned by: MxlSl = Elﬁsxl Z Z Bmm'lmil (ll 1'
wlwre px (:15) is the probability nun58 function of X. (a) 47 points: Express Equation (1) in terms of E[X"]. HINT: The series expansion of an exponential
may be useful. (b) 3 points: Suppose E[X“‘] = 0.8, A“. = 1,2, . . ., ﬁnd the moment generating function MAS). (c) 3 points: Find P(X = (J) and P(X = l). HINT: Part (b) can be helpful. @wnmmeezjg l) Question 3 (Continued) Question 3 (Continued) 1f] ...
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This note was uploaded on 09/27/2011 for the course ECE 2010 taught by Professor Stephenbitar during the Spring '09 term at WPI.
 Spring '09
 StephenBitar

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